Fastest Distributed Consensus Problem on Fusion of Two Star Sensor Networks

Abstract

Finding optimal weights for the problem of Fastest Distributed Consensus on networks with different topologies has been an active area of research for a number of years. Here in this work we present an analytical solution for the problem of Fastest Distributed Consensus for a network formed from fusion of two different symmetric star networks or in other words a network consists of two different symmetric star networks which share the same central node. The solution procedure consists of stratification of associated connectivity graph of network and Semidefinite Programming (SDP), particularly solving the slackness conditions, where the optimal weights are obtained by inductive comparing of the characteristic polynomials initiated by slackness conditions. Some numerical simulations are carried out to investigate the trade-off between the parameters of two fused star networks, namely the length and number of branches.

Gossip algorithm with nonuniform clock distribution: Optimization over classical and quantum networks

“…For ≤ 1 2 (wherẽ≥ ), we have 2 (W Q , ) ≥ 2 (W Q ,̃). Thus, the SDP formulation of (39) for N ≥ d 2 + 1 and 0 < ≤ 1 2 can be written as (40). For 1 2 ≤ < 1 (wherẽ≤ ), we have 2 (W Q , ) ≤ 2 (W Q ,̃).…”

“…The optimization problem (40) and (41) are similar to the fastest continuous-time quantum consensus problem. 28 For both cases of N ≤ d 2 and N ≥ d 2 + 1, as is approaching toward 1 2 (in either directions), the value of optimal convergence rate obtained from (40) and (41), respectively, is getting smaller (ie, faster convergence), where, for = 1 2 , all optimal convergence rates obtained from (40) and (41) are equal. Thus, it can be concluded that the optimal convergence rate as a function of reaches its optimal value for = 1 2 .…”

“…While for 1 2 ≤ < 0, for all values of N and d 2 , the optimal value of the objective function of (39) is equal to 2 (W Q ,̃). Thus, depending on the value of , the FQG problem (38) can be written as the SDP problems (40) and (41), which correspond to the fastest continuous-time quantum consensus problem. 28 This indicates that the resultant optimal quantum gossip algorithm can reach the optimal convergence rate of the continuous-time quantum consensus algorithm.…”

“…In the fourth section, the optimization of the FQG problem (38) is addressed using SDP for nonuniform clock distribution. The results presented in this section include the optimal weights (obtained from (40) and (41)) along with the corresponding optimal clock distribution and the transition probabilities, and the optimal value of SLEM.…”

“…Using relationsẽ T ·ê =ê T ·ẽ = , , the complementary slackness condition 36,40 (F(x) × Z = 0) reduces to…”